to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

A systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering The form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro-differential operator A comprehensive presentation of the inverse scattering method is given and general features of the solution are discussed The relationship of the scattering theory and Backlund transformations is brought out In view of the role of the dispersion relation, the comparatively simple asymptotic states, and the similarity of the method itself to Fourier transforms, this theory can be considered a natural extension of Fourier analysis to nonlinear problems

The Inverse scattering transform fourier analysis for nonlinear problems

Preface Foreword 1. Bilinearization of soliton equations 2. Determinants and pfaffians 3. Structure of soliton equations 4. Backlund transformations Afterword References Index.

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The direct method in soliton theory

Extended tanh-function method and its applications to nonlinear equations

The (G' G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics

An exact solution has been obtained for the Korteweg---de Vries equation for the case of multiple collisions of $N$ solitons with different amplitudes.

Exact Solution of the Korteweg-de Vries Equation for Multiple Collisions of Solitons

Exact N‐envelope‐soliton solutions have been obtained for the following nonlinear wave equation, i∂ψ/∂t + i3α|ψ|2 ∂ψ/∂x + β∂2ψ/∂x2 + iγ∂3ψ/∂x3 + δ|ψ|2ψ = 0, where α, β, γ and δ are real positive constants with the relation αβ = γδ. In one limit of α = γ = 0, the equation reduces to the nonlinear Schrodinger equation which describes a plane self‐focusing and one‐dimensional self‐modulation of waves in nonlinear dispersive media. In another limit, β = δ = 0, the equation for real Ψ, reduces to the modified Korteweg‐de Vries equation. Hence, the solutions reveal the close relation between classical solitons and envelope‐solitons.

Exact envelope‐soliton solutions of a nonlinear wave equation

Soliton solutions of a coupled Korteweg-de Vries equation

A discrete analogue of a generalized Toda equation and its Backlund transformations are obtained. The equation is expressed with the bilinear form as follows \begin{aligned} [Z_{1} \exp (D_{1})+Z_{2} \exp (D_{2})+Z_{3} \exp (D_{3})]f \cdot f=0 \end{aligned} where Z i and D i for i =1, 2, 3, are an arbitrary parameter and a linear combination of the binary operators D t , D x , D y , D n , etc., respectively. The equation is very generic, namely appropriate combinations of parameters give various types of soliton equations including the Korteweg-de Vries equation, Kadomtsev-Petviashvili equation, modified KdV equation, sine-Gordon equation, nonlinear Klein-Gordon equation, Benjamin-Ono equation and various types of discrete analogues of soliton equations.

Discrete Analogue of a Generalized Toda Equation

Exact N‐soliton solutions have been obtained for the nonlinear wave equation ∂2w/∂t2 − ∂2w/∂x2 − 6(∂2w2/x2) − ∂4w/∂x4 = 0 which describes motions of long waves in one‐dimensional nonlinear lattices and in shallow‐water under gravity. The solutions have the same functional form as N‐solition solutions of the Korteweg‐de Vries equation.

Ryogo Hirota

Papers

Exact Solution of the Korteweg-de Vries Equation for Multiple Collisions of Solitons

Exact envelope‐soliton solutions of a nonlinear wave equation

Soliton solutions of a coupled Korteweg-de Vries equation

Discrete Analogue of a Generalized Toda Equation

Exact N‐soliton solutions of the wave equation of long waves in shallow‐water and in nonlinear lattices